Outline

  1. Topic 1
  2. Topic that is a bit long, but it still should fit somehow
  3. Topic 3
  4. Topic 4
  5. Topic 5
  6. Topic 6
  1. Topic 7
  2. Topic 8
  3. Topic 9
  4. Topic 10
  5. Topic that is a bit long, but it still should fit somehow
  6. Topic 12

Outline

  1. Topic 1
  2. Topic that is a bit long, but it still should fit somehow
  3. Topic 3
  4. Topic 4
  5. Topic 5
  6. Topic 6
  7. Topic 7

Topic 1

Not outline, just simple dice

  1. one
  2. two
  3. three

Numbered

  1. Gaming chance
  2. second topic
  3. Third topic
  4. Forth topic

Not numbered

  • Gaming chance
  • second topic
  • Third topic
  • Forth topic

Numbered small

  1. Gaming chance
  2. second topic
  3. Third topic
  4. Forth topic

Not numbered small

  • Gaming chance
  • second topic
  • Third topic
  • Forth topic

Just text

Something like a text here

  • This is my first line and it’s a very long line so that I can check how it looks like when it is printed on the slide and there are other lines below too
  • This is the second line
  • And a third line here

More here

  • Fifth line
  • Sixth line

Code

a <- rnorm(100, 4, 2)
summary(a)
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
 -1.034   2.458   4.025   3.863   5.275   9.488 
hist(a)

Topic 2: a longer topic text that extends over several lines

Testing

Testing

how

fragments work in

reality

Testing 2

Testing

. . .

How

. . .

fragments work

. . .

really

Topic 3: a longer topic text that extends over several lines

Statistics

and the state

statshistory 1660 Hermann Conring (1606–1681) 1660 'statisticum'; political science lectures on European states, quantifying and comparing their finances, population, agriculture, etc. botero Giovanni Botero (c. 1544–1617) 'Della ragion di Stato' ~ 'Reason of State' 1672 Helenus Politanus (pse.) 1672 'Microscopium Statisticum' ghilini Girolamo Ghilini (1589–1668) 'Ristretto della civile, politica, statistica e militare scienza' 1748 Gottfried Achenwall (1719–1772) 1748 'Vorbereitung zur Staatswissenschaft'; 'Statistik' 1660->1672 sittewald Philander Von Sittewald (1601–1669) 'statista' ~ 'someone versed in the knowledge of the state' 1725 Martin Schmeitzel (1679–1747) 1725 'collegium statisticum' ~ 'council of state' 1660->1725 1725->1748 petty William Petty (1623-1687) 1672 'political arithmetic'; 'Essays in Political Arithmetick and Political Survey or Anatomy of Ireland' suss Johann Peter Süssmilch (1707-1767) 1761-1762 'The Divine order in the changes in the human sex from birth, death and reproduction of the same' graunt John Graunt (1620–1674) 1663 'Natural and Political Observations Made upon the Bills of Mortality'

Statistics

and probability





Statistics as the mathematical science of using probability to describe uncertainty





Topic 4: a longer topic text that extends over several lines

Gaming chance

  1. We may never know when humans started playing games of chance, but archaeological findings suggest it was a rather long time ago
  2. During the the First Dynasty in Egypt (c. 3500 B.C.) variants of a game involving astragali (small bones in the ankle of an animal) were already documented
  3. One of the chief games may have been the simple one of throwing four astragali together and noting which sides fell uppermost

Ālea iacta est

  • The six-sided die we know today may have been obtained from the astragalus by grinding it down until it formed a rough cube
  • Dice became common in the Ptolemaic dynasty (300 to 30 B.C.)
  • There is evidence that dice were used for divination rites in this period - one carried the sacred symbols of Osiris, Horus, Isis, Nebhat, Hathor and Horhudet engraved on its six sides
  • In Roman times, rule by divination attained great proportions; Emperors Septimius Severus (Emperor A.D. 193-211) and Diocletian (Emperor AD. 284-305) were notorious for their reliance on the whims of the gods

Fat chance

He threw four knucklebones on to the table and committed his hopes to the throw. If he threw well, particularly if he obtained the image of the goddess herself, no two showing the same number, he adored the goddess, and was in high hopes of gratifying his passion; if he threw badly, as usually happens, and got an unlucky combination, he called down imprecations on all Cnidos, and was as much overcome by grief as if he had suffered some personal loss.

Lucian of Samosata (c. 125 – 180), writing in his trademark satirical style about a young man who fell in love with Praxiteles’s Aphrodite of Knidos; cited in F. N. David (1955:8)

Chance with limitations

  • Dice were sometimes faked. Sometimes numbers were left off or duplicated; hollow dice have been found dating from Roman time
  • Dice were also imperfect; a fair die was the exception rather than the rule
  • Experiment by F. N. David using three dice from the British Museum:

From chance to probability

  • Until 18th century people had mostly used probability to solve problems about dice throwing and other games of chance
  • Jacob (Jacques/James) Bernoulli (1654/1655-1705), a Swiss mathematician trained as a theologian and ordained as a minister of the Reformed church in Basel, began asking questions about probabilistic inference instead
  • His work focused on the mathematics of uncertainty - what he came to call stochastics (from the Greek word \(στόχος\) [stókhos] meaning to aim or “guess’)
  • Ars Conjectandi (The Art of Conjecturing) - published posthumously in 1713

Inferential questions

Suppose you are presented with a large urn full of tiny white and black pebbles, in a ratio that’s unknown to you. You begin selecting pebbles from the urn and recording their colors, black or white. How do you use these results to make a guess about the ratio of pebble colors in the urn as a whole?

  • Bernoulli’s solution: if you take a large enough sample, you can be very sure, to within a small margin of absolute certainty, that the proportion of white pebbles you observe in the sample is close to the proportion of white pebbles in the urn.
  • A first version of the Law of Large Numbers

Large numbers

  • Bernoulli’s solution, more technically:
    For any given \(\epsilon\) > 0 and any \(s\) > 0, there is a sample size \(n\) such that, with \(w\) being the number of white pebbles counted in the sample and \(f\) being the true fraction of white pebbles in the urn, the probability of \(w/n\) falling between \(f − \epsilon\) and \(f + \epsilon\) is greater than \(1 − s\).

  • the fraction \(w/n\) is the ratio of white to total pebbles we observe in our sample

  • \(\epsilon\) (epsilon) captures the fact that we may not see the true urn ratio exactly thanks to random variation in the sample; larger samples help assure that we get closer to the true value, but uncertainty alwa ys remains

  • \(s\) reflects just how sure we want to be; for example, set \(s\) = 0.01 and be 99% percent sure.

  • moral certainty as distinct from absolute certainty of the kind logical deduction provides

Topic 6: a longer topic text that extends over several lines

Final slides

References